Question

Marginal Cost and Marginal Benefit

1. Suppose you are given a benefit equation as B(t) = -3t² + 15t + 160 and cost equation as C(t) = t² -5t + 60, where t = time in hours. Find a NB equation.

b. Solve for MB, MC, and find a MNB expression.

c. Find the optimal amount of time you should spend on the activity and the value of NB associated with that optimal time.

d. If you were given the following tables of benefits and costs for an activity, find the optimal level of the activity. To do this find, MB, MC, NB, and MNB. Explain why it is the optimal amount.

Equation used to solve for column values
Q	Benefits	Costs	NB	MB	MC	MNB
1	60	20
2	110	60
3	150	90
4	180	110
5	200	120
6	210	140
7	210	170
8	200	210

Answer 1

a)Net benefit is the difference between total benefit and total cost.

So ,NB(t)=B(t) -C(t)

-3t²+15t+160 -(t²-5t+60)

-3t²+15t+160-t²+5t-60

NB(t)=-4t​​​​²+20t+100

b)marginal benefit (MB) is addition to total benefit----:

So ,MB equation is derivative of B(t)

Where, B(t)=-3t²+15t+160

  MB=-6t+15

similarly,marginal cost ( MC) is addition to total cost---:

So,MC equation is derivative of C(t)

Where,C(t)=t²-5t+60

  MC= 2t-5

now, MNB that is marginal net benefit is the difference between marginal benefit (MB)and marginal cost(MC)----:

MNB=MB-MC

-6t+15-(2t-5)

So,

MNB=-8t+20

c)Optimal amount of time ,one should spend on an activity ,when,

MB=MC

-6t+15=2t-5

Equating and solving both equation ,we get,

t= 2•5 hours

Optimal time =2•5 hours

value of net benefit when t=2•5,

NB=-4t²+20t+100

Substituting value of t, we get-----:

-4(2•5)²+20(2•5)+100

-25+50+100=125

Value of net benefit=125

d)

Q	Benefit (B)	cost (C)	NB (B)-(C)	MB	MC	MNB MB-MC
1	60	20	40	---	----	----
2	110	60	50	50	40	10
3	150	90	60	40	30	10
4	180	110	70	30	20	10
5	200	120	80	20	10	10
6	210	140	70	10	20	-10
7	210	170	40	0	30	-30
8	200	210	-10	-10	40	-50